Course MT3818 Topics in Geometry

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## Exercises 9

1. Two triangles A1B1C1 and A2B2C2 in R3 are in perspective from O if A1A2 , B1B2 and C1C2 meet at O. The intersection of the planes A1B1C1 and A2B2C2 is called the axis of perspective.
Prove that for two such triangles the four axes of perspective of the pairs A1B1C1 , A2B2C2 : A2B1C1 , A1B2C2 : A1B2C1 , A2B1C2 : A1B1C2 , A2B2C1 are coplanar.
[Project the axis of the pair A1B1C1 , A2B2C2 and the point O to the plane at infinity in RP3 and then look at what this does to the picture.]

2. Show that the elements of PGL(2, R) which map a line to itself form a subgroup isomorphic to the Affine group A(R2).
Deduce that one may recover Affine Geometry from Projective Geometry by considering only those maps which take the line at infinity to itself.

3. Deduce a result from Pascal's Mystic Hexagram Theorem by making consecutive vertices coincide.
Use Brianchon's theorem to deduce a property of a triangle which circumscribes a conic. (i.e. the three sides of the triangle are tangents to the conic.)
In the case when the conic is a circle, give an independent proof of this.

4. Given five points on a conic, use Pascal's theorem to construct any number of other points on the conic.

5. A circle in the elliptic plane is a set { y | d(x, y) = r } for a fixed xRP2 and r > 0.
By drawing the corresponding curves on the 2-sphere S2 find two circles in the elliptic plane which intersect in four points.

6. If an elliptic triangle has angles 0 < α β γ then prove that β + γ < π + α.
For which integers p, q, r ≥ 2 is there an elliptic triangle with angles π/p, π/q, π/r ?

7. In what range do the angles of a regular p-gon in the elliptic plane lie?
Which regular p-gons can be used to tile the elliptic plane?

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JOC March 2003